Deputy Prime Minister Angela Rayner has dismissed an SNP jibe over the row over Labour Party volunteers campaigning in the US presidential election.

Donald Trump’s campaign has filed a complaint alleging Labour has broken US election rules on foreign interference by sending activists to campaign for his Democratic Party opponent, Kamala Harris.

Rayner told MPs that “people in their own time often go on campaigns”, adding that “it happens in all political parties”, during Prime Minister’s Questions.

She was standing in for Sir Keir Starmer, who has also played down the row while on his way to Samoa for the Commonwealth Heads of Government Meeting, which King Charles will open on Friday.

Rayner was responding to a question from Stephen Flynn, leader of the SNP at Westminster, who invited her to join him “in applauding the brave Labour staff members who travelled across the Atlantic to campaign against Donald Trump?”

The deputy PM replied: “People in their own time often go and campaign, and that’s what we’ve seen.

“It happens in all political parties, people go and campaign and they do what they want to do with their own time, with their own money.”

The row over campaigning was sparked by a now-deleted social media post from Labour’s head of operations, Sofia Patel, that she had about 100 current and former party staff heading to America before polling day.

The LinkedIn post said she had “10 spots available” for anyone willing to travel to North Carolina to campaign for Harris, adding “we will sort your housing”.

Foreign nationals are permitted to volunteers in political campaigns in the US as long as they are not compensated, according to Federal Election Commission rules.

Labour Party sources insist no one has done anything wrong, but there is concern about whether the row could impact the so-called “special relationship” between the UK and US should Trump win the election on 5 November.

The Trump campaign complaint to the Federal Election Commission flags that senior Labour Party staff attended the Democratic convention in Chicago and met Harris’s campaign team, naming Morgan McSweeney, the prime minister’s chief of staff, and Matthew Doyle, Downing Street director of communications.

Deborah Mattinson, Sir Keir’s former director of strategy, was also named as someone who travelled to Washington in September to brief Harris’s campaign on Labour’s election-winning approach.

It is understood from Labour officials that Labour met McSweeney’s costs, while Doyle was hosted by the Progressive Policy Institute, a US think tank.

But the officials said it would be wrong to suggest either man had advised or assisted the Harris campaign, adding that Labour sends a delegation to each Democratic convention.

Officials also pointed out that Mattinson left the Labour party staff after July’s general election in the UK.

Sir Keir briefly addressed the issue during his plane journey, telling reporters: “The Labour Party has volunteers, [they] have gone over pretty much every election.

“They’re doing it in their spare time. They’re doing it as volunteers. They’re staying I think with other volunteers over there.”

He denied the row would impact his relationship with Trump, reminding reporters the two had dinner together at Trump Tower in New York last month.

The Prime Minister’s deputy spokesperson stressed that the UK would always have “a deep and strong relationship with the US as our closest ally” whoever won the election.

She was not aware of any plans for government ministers to speak to Trump’s campaign team, but Sir Keir and Trump had discussed “the long-standing friendship” between the two nations during their New York dinner, she said.

“It is a special relationship which has endured for over a century with leaders of all political stripes, and that will always be the case,” she added.

Defence Secretary John Healey suggested the Trump campaign was “creating controversy” ahead of the presidential election.

However, Nigel Farage, the Reform UK leader and Trump supporter, told the BBC he believed the wording of the LinkedIn post breached US election law, saying the rules were “very, very clear”.

Farage, who has travelled to the US to support his friend on multiple occasions, said: “The ad didn’t say you’ll be going in your own free time, didn’t say you’ll have to pay your own air fare, which at the moment, by the way, are very, very expensive, it said you’re going to have free accommodation.

“If you look at the wording of that advert there is little doubt that is against American election law.”

Filings with the US Department of Justice reveal that Farage himself has received help three times since his election as an MP in July from a PR firm based in Pennsylvania, including with hotel costs on a trip to the USA.

The company, Capital HQ LLC, is run by Alexandra Preate, who used to be an aide to Trump’s former strategist Steve Bannon.

The filings show the firm paid more than $3,500 (£2,700) for Farage’s accommodation at the Republican National Convention in Milwaukee in July.

It also helped with his “perception management”, as well as public relations, travel and logistics.

Capital HQ disclosed it had also assisted with his appearance on a Fox News business programme in July, in which he was outspoken in support of Trump’s election prospects.

It also spoke to him in August about an “upcoming trip to the United States”.

The filings were made under US laws requiring anyone who acts on behalf of foreign “principals” engaged in political activity, such as seeking to influence policy or public opinion there, to register with the federal government.

The Clacton MP has declared £32,836 he separately received from a donor for flights and accommodation for a visit to the United States in July.

Asked about this on BBC’s Politics Live programme, Farage said he “hopped on somebody else’s plane, they gave me a free lift”, adding that he had declared it and “didn’t campaign at all”.

“I went in a purely personal capacity to offer my support after the first assassination attempt… as a friend of his and the family”.